( 569 ) 
then ensues, if we represent the radii vectores M'M and J" M of M 
with respect to the fixed points JM’ and M" by w and », that 
“u—v=+(r'—r"). So the locus of the centres of the spherical 
spaces Sp,(M,r) of the system Sy,—2 is the figure of revolution, 
which is generated when the hyperbola H with M' and M" as foci 
and + (r'—r") as half real axis rotates round m in the space S,—1 
through SS, and O'. And because each spherical space Sp, of Sy, 
touches the spherical spaces of Sy, having the vertices of the 
hyperbola H as centres, those vertices of the hyperbola H/ are reversely 
the foci of the ellipse £. Thus the theorem holds good: 
“The spherical spaces Sp„ touching 2 spherical spaces Spn given 
arbitrarily in S, form 2”-! singular infinite series. The spherical 
spaces of any of those series are connected by this that they intersect 
a definite spherical space Sp',® at right angles and that their centres 
lie on a definite conic (A); the determining figures, the spherical 
space Sp’, and the conic (A), change from series to series. To each 
series corresponds as envelope of its spherical spaces a definite curved 
space of order four, the n-dimensional eyclid of Dupin. And if we 
confine ourselves to a single series, the system of n-given spherical 
spaces Sp, can be extended to an n—2-fold infinite series of spherical 
spaces Sp, connected by the fact that they cut another spherical 
space Sp,\° at right angles and that their centres are situated on the 
surface of a figure of revolution generated by the rotation of a conic 
(K). These two conies (K) and (X’') lie in mutually perpendicular 
planes in such a way that the foci of one are vertices of the other 
and reversely.” 
7. The inversion applied becomes impossible within the region of 
reality when the common power of the radical centre O of the n 
given spherical spaces Sp, with respect to those spherical spaces is 
negative. In this case before inverting we can diminish the radii of 
the m given spherical spaces by a common quantity in such a way 
that the radius of one of those spherical spaces disappears. Then the 
power of the radical centre O of the new spheres is certainly positive. 
By operating now with the new system and after that, when the 
system Sy, has been found, by adding the assumed quantity to the 
radii of the spherical spaces of Sy,, we arrive at the desired aim. 
As is evident we can even augment the radii of some of the 
given spherical spaces by the radius of the spherical space that is 
to become a spherical space reduced to a point if only the series of 
the tangent spherical spaces is chosen so as to correspond to this. 
39 
Proceedings Royal Acad. Amsterdam. Vol. VII. 
